Yesterday while learning new integration techniques I stumbled upon the one where we can replace sinx as Img(e^1x) and tried using this for many types of questions. But somehow it didn't seem to work with questions with a log for example:
Image **I meant (pi^2)/8
So could you please correct me with the mistake being made here ?
It is true that $\sin x = \operatorname{Im} e^{ix} $ for real $x$. But it is not true that $\ln(\sin x) = \operatorname{Im} \ln(e^{ix})$.
In fact: $$ \ln(\sin x) = \ln(\operatorname{Im} e^{ix}) $$ but in general $$ \ln(\operatorname{Im} e^{ix}) \ne \operatorname{Im} \ln(e^{ix}) $$
I wrote $\operatorname{Im} $ for "imaginary part".